TABLE OF CONTENTS Subsections: A. Infinite-dimensional topological vector spaces, 569; B. Measures on locally compact spaces, 572; C. Haar measures, 578; D. Measures on Hausdorff spaces, 581; E. Measures on infinite-dimensional vector spaces, 583.
In this Section, we summarize some relevant material on measures on locally compact spaces, locally compact groups, Hausdorff spaces, and infinite-dimensional topological vector spaces.
There are several standard topologies introduced on an infinite-dimensional (complex or real) vector space and its dual [374]. Topological vector spaces throughout the book are assumed to be locally convex. Unless otherwise stated, by the dual V' of a topological vector space V is meant its topological dual, i.e., the space of continuous linear maps of V ? ?.
Let us note that a topology on a vector space V is often determined by a set of seminorms. A non-negative real function p on V is called a seminorm if it satisfies the conditions
A seminorm p for which p( x) = 0 implies x = 0 is called a norm. Given any set { p i} i ? I of seminorms on a vector space V, there is the coarsest topology on V compatible with the algebraic structure such that all seminorms Pi are continuous. It is a locally convex topology whose base of closed neighborhoods consists of the set
Let
